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Unoccupied Topological States on Bismuth Chalcogenides

D. Niesner,1 Th. Fauster,1 S. V. Eremeev,2, 3 T. V. Menshchikova,3

Yu. M. Koroteev,2, 3 A. P. Protogenov,4, 5 E. V. Chulkov,5, 6 O. E.

Tereshchenko,7 K. A. Kokh,8 O. Alekperov,9 A. Nadjafov,9 and N. Mamedov9

1Lehrstuhl fur Festkorperphysik, Universitat

Erlangen-Nurnberg, D-91058 Erlangen, Germany

2Institute of Strength Physics and Materials Science, 634021, Tomsk, Russia

3Tomsk State University, 634050 Tomsk, Russia

4Institute of Applied Physics, Nizhny Novgorod, 603950, Russia

5Donostia International Physics Center (DIPC),

20018 San Sebastian/Donostia, Basque Country, Spain

6Departamento de Fısica de Materiales UPV/EHU and Centro de

Fısica de Materiales CFM and Centro Mixto CSIC-UPV/EHU,

20080 San Sebastian/Donostia, Basque Country, Spain

7Institute of Semiconductor Physics, Novosibirsk, 630090 Russia

8Institute of Geology and Mineralogy, Novosibirsk, 630090 Russia

9Institute of Physics, Azerbaijan National Academy of Sciences, AZ1143 Baku, Azerbaijan

Abstract

The unoccupied part of the band structure of topological insulators Bi2TexSe3−x (x = 0, 2, 3) is

studied by angle-resolved two-photon photoemission and density functional theory. For all surfaces

linearly-dispersing surface states are found at the center of the surface Brillouin zone at energies

around 1.3 eV above the Fermi level. Theoretical analysis shows that this feature appears in a spin-

orbit-interaction induced and inverted local energy gap. This inversion is insensitive to variation of

electronic and structural parameters in Bi2Se3 and Bi2Te2Se. In Bi2Te3 small structural variations

can change the character of the local energy gap depending on which an unoccupied Dirac state

does or does not exist. Circular dichroism measurements confirm the expected spin texture. From

these findings we assign the observed state to an unoccupied topological surface state.

PACS numbers: 73.20.-r, 79.60.Bm

1

Three-dimensional topological insulators (TIs) are insulators in bulk and metals at the

surface. Metallic character of a TI surface is determined by a massless Dirac state that

crosses the Fermi level EF [1–3]. This spin-polarized linearly-dispersing surface state arises

from a symmetry inversion of the bulk bands at band gap edges owing to strong spin-orbit

interaction (SOI). Various new phenomena in TIs were predicted like dissipationless spin

transport [4], formation of Majorana fermions in the presence of superconductors [5] and

magnetic monopoles [6].

Most of these phenomena deal with low-energy electronic excitations and transport. Little

experimental work has been performed on excited electronic states and their dynamics. The

best studied three-dimensional topological insulators at present are bismuth chalcogenides.

The equilibrium band structure of these TI’s is well understood [3, 7–13] and the spin

texture of the topological surface state (TSS) was observed both indirectly by use of circular

dichroism (CD) [14] and directly by spin-resolved experiments [8, 15].

The materials are usually intrinsically n-doped, which complicates direct spectroscopic

access to excited electrons in a TSS. Besides an earlier inverse photoemission investigation

[16] a recent study was performed by Sobota et al. demonstrating that a persistent occu-

pation of the Dirac cone close to the Fermi level in p-doped Bi2Se3 can be supported even

picoseconds after an initial optical excitation [17].

Here we present a different approach by investigating higher-excited TSSs in Bi2TexSe3−x

(x = 0, 2, 3). In the calculated band structures symmetry-inverted band gaps are identified

which support an empty Dirac cone. Experimental proof is given by monochromatic two-

photon photoemission (2PPE). Calculations together with CD measurements reveal the

characteristic spin structure of a topological surface state.

Angle-resolved monochromatic 2PPE and one-photon photoemission (ARPES) exper-

iments were conducted using the third and fourth harmonic (4.65 eV and 6.2 eV photon

energy) of a titanium:sapphire oscillator with a repetition rate of 80MHz and pulse lengths

around 100 fs [18, 19]. In both cases the beam is initially p-polarized with an incidence

angle of 45◦. Circular polarization of the third harmonic necessary for circular dichroism

experiments was obtained using a λ/4-waveplate. Two-dimensional momentum distribution

patterns (MDCs) at constant kinetic energy were recorded using an ellipsoidal “display-

type” analyzer at an energy and angular resolution of 50meV and 3◦, respectively [19, 20].

Angle-resolved spectra for the intensity maps were acquired by a hemispherical analyzer

2

with resolution of 34meV and 1.6◦, respectively [18].

Bi2Te2Se was prepared from presynthesized mixtures of Bi2Te3 and Bi2Se3, which in turn

were prepared from elementary Bi, Te and Se of 99.999% purity. Crystal growth for all three

compounds was done in sealed quartz ampoules coated with a carbon layer. For recrystalliza-

tion we used a vertical variant of the modified Bridgman method [21]. The resulting ingots

consisted of one or several large single-crystalline blocks. Bi2Se3, Bi2Te3 and Bi2Te2Se sam-

ples were naturally n-doped (through the formation of defects) with carrier concentration in

the range of (1−9)×1018 cm−3. Samples were cleaved in vacuum at a pressure < 5×10−6 Pa

and then transferred to ultrahigh vacuum (pressure < 1× 10−8 Pa) where they were cooled

to 90K for measurements. Sample quality and orientation was checked by low-energy elec-

tron diffraction (LEED) showing a sharp threefold pattern. According to LEED patterns

the samples were oriented with the laser beams incident parallel to the ΓM mirror plane or

along the ΓK direction.

Because the description of the unoccupied states is a delicate issue of density functional

theory (DFT) we employ two different computer codes that are based on DFT. The main

part of the electronic structure calculations was performed within the density functional

formalism implemented in VASP [22, 23]. We used the all-electron projector augmented wave

(PAW) [24, 25] basis sets with the generalized gradient approximation of Perdew, Burke, and

Ernzerhof (PBE) [26] to the exchange correlation (XC) potential. For bulk band structure

calculations the local density approximation (LDA) [27] to XC potential was also tested. The

experimental lattice parameters were used for calculation of all the considered compounds

while atom positions within the unit cell were optimized. The Hamiltonian contains scalar

relativistic corrections, and the SOI is taken into account by the second variation method

[28]. To simulate the semi-infinite Bi2TexSe3−x(111) we use a slab composed of 9 quintuple

layers (QLs).

The second approach used for electronic structure calculations is the full-potential lin-

earized augmented plane-wave (FLAPW) method as implemented in the FLEUR code [29]

with PBE for the exchange-correlation potential. Spin-orbit coupling was included in the

self-consistent calculations as described in Ref. [30]. The FLAPW basis has been extended

by conventional local orbitals to treat quite shallow semi-core d-states. Additionally, to

describe high-lying unoccupied states accurately, we have included for each atom one local

orbital per angular momentum up to l = 3. The fundamental quantities are consistently

3

FIG. 1. (Color online) Measured (top row) and calculated (bottom row) data on the band structure

of Bi2Te2Se. Left (right) column shows occupied (unoccupied) states. Shaded regions in the

calculations, (c) and (d), indicate projected bulk bands, whereas red dots indicate an increased

localization within the topmost QL.

reproduced by both methods [31] except for the case Bi2Te3 which will be addressed below

in detail.

To study the effect of dispersion interactions we use the van-der-Waals non-local correla-

tion functional (vdW-DF) as implemented in the VASP code [32]. Both lattice parameters

and internal atomic positions were optimized in this approach.

We start with the ternary tetradymite compound Bi2Te2Se which recently was shown to

4

FIG. 2. (Color online) Close-up of the band gap under investigation for the three materials. No

azimuthal dependence was observed close to Γ and the sample orientation was optimized to access

the Dirac point in experiment.

be a three-dimensional TI by ARPES measurements [33, 34]. The crystal structure of this

compound is obtained by replacing the central Te layer of each QL of Bi2Te3 by a Se layer.

The results of photoemission experiments and DFT calculations on Bi2Te2Se are shown

in Fig. 1. Shaded regions in the calculated band structure depict projected bulk bands,

while red dots indicate an increased localization in the first quintuple Bi2Te2Se layer. From

the ARPES data (Fig. 1(a)) the Dirac cone in the occupied part of the band structure is

found 0.25 eV below EF. Depending on crystal growth conditions values between 0.1 and

5

0.4 eV have been found [33, 34]. In the 2PPE data (Fig. 1(b)) all prominent features can

be attributed to unoccupied bands in the calculations (Fig. 1(d)) and we conclude that the

2PPE process is dominated by intermediate states. The measured bands of high intensity

coincide well with surface resonances in the calculation, giving evidence of a high surface

sensitivity. A similar degree of agreement between theory and experiment is also found for

Bi2Se3 and Bi2Te3 surfaces. The present results are in qualitative agreement with the inverse

photoemission spectra which show three conduction band peaks [16].

In the following we will focus on the projected bulk band gaps at energies above EF.

Figures 2(a) and (d) show a close-up of such a gap for Bi2Te2Se. In fact both 2PPE and

DFT data show that it is bridged by two linearly dispersing bands which cross at the Γ-

point. At this energy it appears in MDCs as a single spot evolving into a circle towards

higher energies (see also Fig. 4(a) and [35]). At energies below the crossing point it turns

into a surface resonance degenerate with bulk bands.

The appearance of the Dirac state in the conduction-band local gap raises the question:

Is this surface state topologically protected? In contrast to the Dirac state located in the

principal energy gap the new massless Dirac state is located in an local energy gap and upon

going along a path between time-reversal invariant momenta (TRIMs) in the Brillouin zone

the gap in the spectrum of bulk states closes and opens.

The origin of the reduction of the contribution made by extended states can be found by

comparing the continuous and lattice versions of the Z2 invariant. Analysis of the relation

(−1)ν0 = (−1)2P3 [36] of the index ν0 [1] in the theory of topological insulators with the

winding number 2P3 does not solve the problem. It is shown in Ref. [37] that the Z2

invariant in the continuous case is alternatively expressed as

D =1

2πi

[∮

∂B−

A−

∫

B−

F

]

mod 2, (1)

where B− = [−π, π] ⊗ [−π, 0] is half of the Brillouin zone, A = Trψ†dψ and F = dA are

the Berry gauge potential and the associated field strength, respectively, and ψ(k) is the

2M(k)-dimensional ground state multiplet. The lattice analog of Eq. (1) is [38]

DL ≡1

2πi

[

∑

k∈∂B−

A(k)−∑

k∈B−

F (k)

]

= −∑

k∈B−

n(k) mod 2 , (2)

6

since∑

k∈B−F (k) =

∑

k∈∂B−A(k) + 2πi

∑

k∈B−n(k). n(k) in this equation are integers and

n(k)mod 2 ∈ Z2 due to the residual U(1) invariance [38].

From Eq. (2) we infer that the reason for cancelation of the contribution made by

bulk spectrum states is the compactness of the lattice gauge theory. The existence of the

second Dirac cone points out the nontrivial value DL = 1mod 2 of the Z2 invariant in this

topologically protected state.

Let us consider now the bulk conduction band of Bi2Te2Se in the energy range of interest.

Without spin-orbit coupling the second and third conduction bands are degenerate along the

ΓZ direction (Fig. 3(a)). Spin-orbit interaction lifts this degeneracy and opens a gap between

these bands (Fig. 3(b)). Thus, the gap supporting the new “X” shaped surface state in the

conduction band originates from SOI. In contrast to the principal Γ-gap edges, which are

composed of inverted Bi and Te states, both the upper and lower bands of the conduction

band Γ-gap are mostly composed of Bi states, however the Se states contribute to these

bands as well. As one can see in Fig. 3(c) in the vicinity of Γ there are two parabolic-like

bands, gapped at points, where they are inverted due to the central Se atom contribution.

This inversion of the local Γ-gap edges in the conduction band along with the presence

at the same energy of the gaps at F and L TRIMs of the bulk Brillouin zone (which are

projected onto the M point of the 2D Brillouin zone) is responsible for the emergence of the

unoccupied Dirac-like surface state at Bi2Te2Se(111).

As one can see in Figs. 2(e) and (f), the FLEUR results show the topological conduction-

band surface state arising in Bi2Se3 and Bi2Te3 that is fully consistent with the outcome of

the experiment. The VASP vdW-DF results are almost the same: they show 10 − 20 meV

smaller unoccupied gap and slightly different position of the surface state band crossing

within this gap. The arising of the unoccupied cone in Bi2Se3 and Bi2Te3, like in Bi2Te2Se,

results from inversion of the QL central atom states of the SOI-induced Γ local gap. The LDA

and PBE calculations performed without taking into account the vdW forces for Bi2Te2Se

and Bi2Se3 give the inverted gap too, while in the case of Bi2Te3 the SOI-induced gap

is not inverted (Fig. 3(d) and Ref. [31]). The dissimilar character of the gap in Bi2Te3

in PBE and LDA results in absence of the unoccupied cone in the surface band structure

(Fig. 3(e)). As mentioned above the different calculations were accompanied by optimization

of the crystal structure. Thus, vdW-DF optimization for all systems under consideration

changes a and c lattice parameter in the range of 0.5 − 1 % and 0.1 − 2 % as compared to

7

FIG. 3. (Color online) Bulk band structure of Bi2Te2Se calculated with VASP (only F − Γ and

Γ − Z directions are shown): (a) without and (b) with SOI included; (c) magnified view of blue

dashed frame marked in panel (b) with the weight of Se states marked in the second and third

conduction bands near the Γ point. Bulk spectrum of Bi2Te3 in the vicinity of the local gap within

vdW-DF, PBE, and LDA approaches (d). 9 QL-slab electronic structure of Bi2Te3 within PBE (e).

Bulk spectrum of Bi2Te3 as calculated by FLEUR code with different experimental parameters:

blue line, Ref. [39]; red line, Ref. [40] (f).

8

experimental values, respectively. At the same time both the full structural optimization

and the relaxation of atomic positions do not change the fractional atomic coordinates more

than 10−3. To determine which factor (XC approximation or structural parameters) is

more important for the change of the gap character we performed calculations of the bulk

electronic structure of Bi2Te3 with two slightly different experimental parameters (Refs. [39]

and [40]) with the same exchange-correlation approximation. As one can see in Fig. 3(f)

this small variation of the crystal structure changes the character of the gap. The latter

means that the emergence of the unoccupied Dirac state in Bi2Te3 can be sensitive to the

sample preparation. As shown in Ref. [41] even a small deviation from the stoichiometric

composition in Bi2Te3 results in the change of the lattice parameters. The data measured in

the present experiment demonstrate relatively weak intensity in the cone region (Fig. 2(c))

which may be related to delicate stability of the unoccupied TSS in Bi2Te3 sample used.

On the other hand the highest intensity observed for Bi2Se3 (Fig. 2(b)) reflects the more

localized character of the TSS which is a consequence of the wider gap in Bi2Se3 as compared

to other materials.

Additional evidence of the topological character of the unoccupied surface state emerges

from its spin structure which can be accessed experimentally by use of CD. Therefore MDCs

were recorded using left- and right-handed circularly polarized light. In Fig. 4(a) and (b) the

sum Iσ++Iσ− and the difference Iσ+−Iσ− of the intensities are shown for Bi2Se3 at an energy

0.3 eV above the Dirac point. These data were obtained for a different crystal than those

in Fig. 2(b) which results in a slightly different energy for the Dirac point. The normalized

asymmetry covers a range of ±12% somewhat smaller than that observed by ARPES for

the occupied Dirac cone where values of 30% for Bi2Se3 [42] and up to 40% for Bi2Te3

[43, 44] have been reported. The lower asymmetry in our experiment might be attributed

to the lower angular and energy resolution than in the ARPES experiments leading to an

increased unpolarized background in the narrow band gap. The two transitions in 2PPE

might also have opposite contributions to the CD pattern reducing the observed asymmetry.

The MDCs show the expected ring and the CD pattern shows the threefold symmetry of the

substrate as observed further away from the Dirac point for the occupied TSS [14, 43, 45]

Figure 4(c) demonstrates the development of the CD pattern over a wide energy range along

the ΓK direction. The dichroism is antisymmetric indicating opposite spin orientation for

the two branches.

9

x

yy

FIG. 4. (Color online) MDCs of the sum (a) and difference (b) of the intensities for left and right

polarized light for an energy of 0.30 eV above the Dirac point for Bi2Se3. (c) Circular dichroism

of unoccupied Dirac cone along the ΓK direction. (d) Calculated spin structure of the unoccupied

cone as represented by spin projections Sx, Sy, and Sz at an energy of ±50 meV with respect to

the crossing point.

10

Figure 4(d) presents the calculated orientation of the electron spin within the surface

state. The calculated spin-resolved constant energy contours (CECs) below and above the

crossing point of the unoccupied cone state show an ideal circular shape of the CECs in

agreement with present experiment and in-plane spin polarization (out-of-plane component

Sz is negligibly small) with positive (clockwise) spin helicity in the upper part and negative

helicity in the lower part of the cone, i. e. the same helicity as in the Dirac cone in the

principal energy gap.

In summary, we presented a study of the unoccupied part of the band structure of bis-

muth chalcogenides. Empty Dirac cones on Bi2Se3, Bi2Te2Se, and Bi2Te3 are identified in

both DFT calculations and 2PPE experiments by their linear dispersion and helical spin

structure. Their existence depends on a symmetry-inverted band gap. In the case of Bi2Te3

this inversion is sensitive to small variations of the structural parameters that can cause

dependence of the empty Dirac cone in this system on sample growth conditions. The origin

of the reduction of the bulk non-TRIM states contribution has been found using the Z2 in-

variant. The observed unoccupied TSS opens a new route to measurements on excitations in

and between topological states. For TI-semiconductor junctions the unoccupied TSS might

pave a way to inject and control spin-polarized currents in the semiconductor conduction

band. The image-potential state seen at an energy of 4.5 eV in Fig. 1(b) corresponds to the

situation discussed for magnetic monopoles [6]. Time-resolved photoemission experiments

on image-potential states and TSSs are in progress.

We thank Philip Hofmann for supplying additional topological insulator samples showing

similar results. SVE thanks Dirk Lamoen and I. A. Nechaev for fruitful discussions.

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13